Hyperbolas

The "basic" hyperbola is the function

y
=
1
x

This function has a graph which consists of two disjoint parts. Note that
0
is not in the
domain
. Note also that the function approaches
0
asymptotically
as
x
grows infinitely large (or infinitely negative), and that it approaches infinity as
x
approaches
0
from the positive side, negative infinity as
x
approaches
0
from the negative side.

The
center
of the hyperbola is the point of rotational symmetry. In the above example, the center is
(
0
,
0
)
.

The graph of the rational function

y
=
a
x
−
h
+
k

is a hyperbola whose center is
(
h
,
k
)
. The constant
a
controls the "steepness" with which the graph approaches the
asymptotes
.

A hyperbola can also be defined as a conic section obtained by the intersection of a double cone with a plane that is intersects both pieces of the cone without intersecting the axis.